Brief Notes
● A Serre Sylow Problem, written after the question was posed to me by Jeremy Booher; the question deals with lifting Sylow subgroups of a quotient G/H to Sylow subgroups of G.
● Inversion is Smooth. Often Lie Groups are defined as manifolds with a topological group structure, such that multiplication and inversion are smooth. This note shows that the smoothness of inversion follows from the rest of the definition.
● The Nullstellensatz. A quick (and essentially standard) proof of the Nullstellensatz, using the Rabinowitz trick.
● Dual Cantor-Schroeder-Bernstein for Rings. Answers a question posed by Sam Lichtenstein; essentially, can a Noetherian ring be isomorphic to a proper quotient of itself?
● Notes on Linear Algebraic Groups for Quals. Gives techniques and examples for analyzing linear algebraic groups over finite fields, with an aim towards solving related problems on the old Stanford Algebra Quals.
● Closed Subgroups of Lie Groups are Lie Subgroups. Shows that subgroups of Lie groups which are closed as topological spaces are closed Lie subgroups, i.e. they are closed submanifolds.
● Kernels of Surjective Bundle Maps need not be Bundles. Answers an offhand question of Ravi Vakil--is the kernel of a surjective map of (infinitely generated) locally free modules locally free? Though the answer is "yes" in the finitely generated case, it is false in general. I give a counterexample with *free* modules.